Quantum phase transitions and disorder:Griffiths singularities, infinite randomness, and smearing

Thomas VojtaDepartment of Physics, Missouri University of Science and Technology

• Phase transitions and quantum phase transitions• Effects of impurities and defects: the common lore• Rare regions, Griffiths singularities and smearing• Experiments in Ni1−xVx and Sr1−xCaxRuO3

• Classification of weakly disordered phase transitions

Sao Carlos, Aug 6, 2014

Acknowledgements

At Missouri S&T:

Rastko Jose Chetan David Fawaz Manal Hatem

Sknepnek Hoyos Kotabage Nozadze Hrahsheh Al Ali Barghathi

PhD ’04 Postdoc ’07 PhD ’11 PhD ’13 PhD ’13 PhD ’13

Experimental Collaborators:

Almut Schroeder Istvan Kezsmarki

(Kent State) (TU Budapest)

Phase transitions: the basics

Phase transition:singularity in thermodynamicquantities

occurs in macroscopic (infinite)systems

1st order transition:phase coexistence, latent heat,finite correlations

solid

liquid

gaseoustripel point

critical point

100 200 300 400 500 600 70010 2

10 3

10 4

10 5

10 6

10 7

10 8

10 9

T(K)

Tc=647 K

Pc=2.2*108 Pa

Tt=273 K

Pt=6000 Pa

Continuous transition:no phase coexistence, no latent heat, critical behavior:

diverging correlation length ξ ∼ |T − Tc|−ν and time ξτ ∼ ξz ∼ |T − Tc|−νz

power-laws in thermodynamic observables: ∆ρ ∼ |T − Tc|β, κ ∼ |T − Tc|−γ

critical exponents are universal = independent of microscopic details

Quantum phase transitions

occur at zero temperature as function of pressure, magnetic field, chemicalcomposition, ...

driven by quantum rather than thermal fluctuations

paramagnet

ferromagnet

0 2 4 6 8B (T)

0.0

0.5

1.0

1.5

2.0

Bc

LiHoF4

phase diagram of LiHoF4 (Bitko et al. 96)

Transverse-field Ising model

H = −J∑⟨i,j⟩

σzi σzj − h

∑i

σxi

transverse magnetic field induces spinflips via σx = σ+ + σ−

transverse field suppresses magneticorder

Imaginary time and quantum to classical mapping

Classical partition function: statics and dynamics decouple

Z =∫dpdq e−βH(p,q) =

∫dp e−βT (p)

∫dq e−βU(q) ∼

∫dq e−βU(q)

Quantum partition function: statics and dynamics coupled

Z = Tre−βH = limN→∞(e−βT/Ne−βU/N)N =∫D[q(τ)] eS[q(τ)]

imaginary time τ acts as additional dimensionat T = 0, the extension in this direction becomes infinite

Caveats:• mapping holds for thermodynamics only• resulting classical system can be unusual and anisotropic (z = 1)• if quantum action is not real, extra complications may arise, e.g., Berry phases

• Phase transitions and quantum phase transitions

• Effects of impurities and defects: the common lore

• Rare regions, Griffiths singularities and smearing

• Experiments in Ni1−xVx and Sr1−xCaxRuO3

• Classification of weakly disordered phase transitions

Phase transitions and (weak) disorder

Real systems always containimpurities and other imperfections

Weak (random-Tc) disorder:

spatial variation of coupling strength butno change in character of the ordered phase

Will the phase transition remain sharp or become smeared?

Will the critical behavior change?

How important are rare strong disorder fluctuations?

Harris criterion

Harris criterion:variation of average local Tc(i) between correlation volumes must be smaller thandistance from global Tc

variation of average Tc(i) in volume ξd

∆Tc(i) ∼ ξ−d/2

distance from global critical pointT − Tc ∼ ξ−1/ν

∆Tc(i) < T − Tc ⇒ dν > 2

ξ

+TC(1),

+TC(4),

+TC(2),

+TC(3),

• if clean critical point fulfills Harris criterion ⇒ stable against disorder• inhomogeneities vanish at large length scales• macroscopic observables are self-averaging• example: 3D classical Heisenberg magnet: ν = 0.711

Finite-disorder critical points

if critical point violates Harris criterion ⇒ unstable against disorder

Common lore:

• new, different critical point which fulfills dν > 2• inhomogeneities finite at all length scales (”finite disorder”)• macroscopic observables not self-averaging• example: 3D classical Ising magnet: clean ν = 0.627 ⇒ dirty ν = 0.684

Distribution of critical susceptibilitiesof 3D dilute Ising model(Wiseman + Domany 98)

0.0 0.5 1.0 1.5 2.0 2.5χ(Tc,L)/[χ(Tc,L)]

0.00

0.02

0.04

0.06

0.08

0.10

freq

uenc

y10.0 100.0

L

0.10

0.12

0.14

0.16

0.18

0.20

Rχ

Disorder and quantum phase transitions

Disorder is quenched:

• impurities are time-independent

• disorder is perfectly correlated in imaginarytime direction

⇒ correlations increase the effects of disorder(”it is harder to average out fluctuations”)

x

τ

Disorder generically has stronger effects on quantum phase transitionsthan on classical transitions

Random transverse-field Ising model

H = −∑⟨i,j⟩

Jijσzi σ

zj−

∑i

hiσxi

nearest neighbor interactions Jij and transverse fields hi both random

Strong-disorder renormalization group: exact solution in 1+1 dimensions:

• Ma, Dasgupta, Hu (1979), Fisher (1992, 1995)• in each step, integrate out largest of all Jij, hi• cluster aggregation/annihilation process• exact in the limit of large disorder

J=J2 J3/h3

h5h4h3h2h1

J1

J1 J4

J3 J4J2

~

Infinite-disorder critical point:

• under renormalization the disorder increases without limit• relative width of the distributions of Jij, hi diverges

Infinite-disorder critical point

• extremely slow dynamics log ξτ ∼ ξµ (activated scaling)• distributions of macroscopic observables become infinitely broad• average and typical values drastically differentcorrelations: Gav ∼ r−η , − logGtyp ∼ rψ

• averages dominated by rare events

Probability distribution of

end-to-end correlations in

a random quantum Ising

chain

(Fisher + Young 98)

• Phase transitions and quantum phase transitions

• Effects of impurities and defects: the common lore

• Rare regions, Griffiths singularities and smearing

• Experiments in Ni1−xVx and Sr1−xCaxRuO3

• Classification of weakly disordered phase transitions

Rare regions in a classical dilute ferromagnet

• critical temperature Tc is reducedcompared to clean value Tc0

• for Tc < T < Tc0:no global order but local order on rareregions devoid of impurities

• probability: w(L) ∼ e−cLd:

rare regions cannot order statically but act as large superspins

⇒ very slow dynamics, large contribution to thermodynamics

Griffiths region or Griffiths “phase”

Griffiths:

rare regions lead to singular free energyeverywhere in the interval Tc < T < Tc0

Rare region susceptibility:

• susceptibility of single RR: χ . L2d/T• sum over all RRs:

χRR ∼∫dL e−cL

dL2d

• essential singularity• large regions make negligible contribution

In generic classical systems:

Thermodynamic Griffiths effects are weak and essentially unobservable

Long-time dynamics can be dominated by rare regions

Quantum Griffiths effects

Quantum phase transitions:

• rare regions are finite in space butinfinite in imaginary time

• fluctuations even slower than inclassical case

Griffiths singularities enhanced

t

rare region at a quantum phase transition

Random quantum Ising systems:

• susceptibility of rare region: χloc ∼ ∆−1 ∼ eaLd

χRR ∼∫dL e−cL

deaL

dcan diverge inside Griffiths region

• power-law quantum Griffiths singularities

susceptibility: χRR ∼ T d/z′−1

specific heat: CRR ∼ T d/z′

z′ is continuously varying Griffiths dynamical exponent, diverges at criticality

Smeared phase transitions

Randomly layered classical magnet:

• layers of two different ferromagneticmaterials grown in random order

• rare regions are thick slabs of the materialwith higher Tc

• rare regions are two-dimensional

• two-dimensional (Ising) magnets have truephase transition

⇒ global magnetization develops gradually as rare regions order independently

⇒ no Griffiths region

global phase transition is smeared by disorder

T.V., Phys. Rev. Lett 90, 107202 (2003); J. Phys. A 36, 10921 (2003)

Isolated islands – Optimal fluctuation arguments

• probability for finding region of size L devoid of weak planes: w ∼ e−cLd⊥

• region has transition at temperature Tc(L) < T 0c (T 0

c = higher of the two bulk Tc)

• finite size scaling: |Tc(L)− T 0c | ∼ L−ϕ (ϕ = clean shift exponent)

probability for finding a region whichbecomes critical at Tc:

w(tc) ∼ exp(−B |Tc − T 0c |−d⊥/ϕ)

total magnetization at temperature T :sum over all rare regions with Tc > T :

m(t) ∼ exp(−B |T−T 0c |−d⊥/ϕ) (T → T 0

c−)

Dissipative random transverse-field Ising model

H = −∑i

Jiσzi σ

zi+1−

∑i

hiσxi+

∑i,n

σzi λi,n(a†i,n + ai,n) +

∑i,n

νi,na†i,nai,n

• nearest neighbor interactions Ji and transverse fields hi both random• bath oscillators a†i,n, ai,n have Ohmic spectral density

E(ω) = π∑n λ

2i,nδ(ω − νi,n) = 2παωe−ω/ωc

• damping due to baths leads to long-range interaction in time: ∼ 1/(τ − τ ′)2

1D Ising model with 1/r2 interaction is known to have an ordered phase

⇒ isolated rare region can develop a static magnetization, i.e.,large islands do not tunnel

⇒ quantum Griffiths behavior does not existmagnetization develops gradually on independent rare regions

quantum phase transition is smeared by disorder

J.A. Hoyos and T.V., PRL 100, 240601 (2008)

Universality of the smearing scenario

Condition for disorder-induced smearing:

isolated rare region can develop a static order parameter⇒ rare region has to be above lower critical dimension

Examples:

• quantum phase transitions in dissipative quantum magnets(disorder correlations in imaginary time + long-range interaction 1/τ2)

• classical Ising magnets with planar defects(disorder correlations in 2 dimensions)

• classical non-equilibrium phase transitions in the directed percolation universalityclass with extended defects(disorder correlations in at least one dimension)

Disorder-induced smearing of a phase transition is aubiquitous phenomenon

• Phase transitions and quantum phase transitions

• Effects of impurities and defects: the common lore

• Rare regions, Griffiths singularities and smearing

• Experiments in Ni1−xVx and Sr1−xCaxRuO3

• Classification of weakly disordered phase transitions

Rare regions in metallic quantum magnets

Gaussian propagator:

G−10 (q, ωn) =

{t+ qd−1 + |ωn|/|q| ferromagnett+ q2 + |ωn| antiferromagnet

magnetic fluctuations are damped due to coupling to electrons

in imaginary time: long-range power-law interaction ∼ 1/(τ − τ ′)2

Consider single rare region:

Itinerant Ising magnets: rare region can order by itself(1D Ising model with 1/r2 interaction has an ordered phase)⇒ global phase transition is smeared

Itinerant Heisenberg magnets: rare region is at the lower critical dimension(1D Heisenberg model with 1/r2 interaction does NOT have an ordered phase)⇒ strong power law quantum Griffiths effects

T.V., Phys. Rev. Lett 90, 107202 (2003); T.V. + J. Schmalian, Phys. Rev. B 72, 045438 (2005)

Phase diagram of Ni1−xVx

0.1

1

10

100

1000

0 2 4 6 8 10 12 14 16

Tc

Tmax

Tcross

Tc (

K)

and o

ther

T-s

cale

s

x (%)

Ni1-x

Vx

PM

FM

GP

CG ?

0

0.2

0.4

0.6

0 5 10 15

- 5µB/V

µsat(µ

B)

x(%)

S. Ubaid-Kassis, T. V. and A. Schroeder, Phys. Rev. Lett. 104, 066402 (2010)

A. Schroeder, S. Ubaid-Kassis and T. V., J. Phys. Condens. Matter 23, 094205 (2011)

Quantum Griffiths singularities in Ni1−xVx

0.1

1

10

100

1000

100 1000 104

Mm (

em

u/m

ol)

H (G)

Ni1-x

Vx

T=2K

(b)

0.0001

0.001

0.01

0.1

1

1 10 100

x=11.4%

11.6%

12.07%

12.25%

13.0%

15.0%!

"1+ "

dyn

"m (

em

u/m

ol)

T(K)

H=100G

(a)

0.1

1

10

100

1000

100 1000 104

Mm (

em

u/m

ol)

H (G)

Ni1-x

Vx

T=2K

(b)

0

0.2

0.4

0.6

0.8

1

11 12 13 14 15 16

!loT

!

1-"

!dyn!,

1- "

x (%)

(a)

Ni1-x

Vx

0.01

0.1

101

102

103

104

105

H=500GT=2KT=14K -- T=300K

Mm/H

0.5 (

em

u m

ol-1

G-0

.5)

H/T (G/K)

x=12.25%

(b)

• χ(T ) and m(H) shownonuniversal power lawsabove xc

• Griffiths exponent λ = d/z′

varies systematically

• λ = 1− γ vanishes atcriticality

improved theory:

• includes Landau damping andorder parameter conservation

• χ ∼ 1T exp

[− dz′| lnT |

3/5]

D. Nozadze + T. V., Phys. Rev. B 85,

174202 (2012)

Phase diagram of Sr1−xCaxRuO3

L. Demko, S. Bordacs, T. Vojta, D. Nozadze, F. Hrahsheh, C. Svoboda, B. Dora, H. Yamada, M.

Kawasaki, Y. Tokura and I. Kezsmarki, Phys. Rev. Lett. 108, 185701 (2012)

Composition-tuned smeared phase transitions

1

2

3

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.500.0

0.2

0.4

0.6

0.8

1.0

-150 0 150

0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52

-6

-5

-4

-3

-2

-1

0

1

2

xC

TC(x)

Theory

ln(T

C)

b

M

B [mT]

M [

B/R

u]

Composition, x

a T=2.9K

4.2K

10K

20K

30KI

0.1 B/Ru

0.52

0.49

0.48

0.47

0.46

0.45

0.44

M(x); T=2.9K

M(x); T=4.2K

M(B); T=4.2K

Theory

ln(M

)

Composition, x

Magnetization and Tc in tail:

M,Tc ∼ exp

[−C

(x− x0c)

2−d/ϕ

x(1− x)

]for x → 1:

M,Tc ∼ (1− x)Ldmin

F. Hrahsheh et al., PRB 83, 224402 (2011)

C. Svoboda et al., EPL 97, 20007 (2012)

T

x

Magneticphase

Tc

decreasesexponentially

0 1

power-law tail

Super-PMbehavior

xc

0

• Phase transitions and quantum phase transitions

• Effects of impurities and defects: the common lore

• Rare regions, Griffiths singularities, and smearing

• Experiments in Ni1−xVx and Sr1−xCaxRuO3

• Classification of weakly disordered phase transitions

Disorder at phase transitions: two frameworks

• fate of average disorder strength under coarse graining

• importance of rare regions and strength of Griffiths singularities

Recently:

• general relation between Harris criterion and rare region physicsT.V. + J.A. Hoyos, Phys. Rev. Lett. 112, 075702 (2014), Phys. Rev. E 90, 012139 (2014)

• below d+c , same inequality, dν > 2, governs relevance or irrelevance of disorderand fate of the Griffiths singularities

• above d+c , behavior is even richer

• relevance of rare regions depends on inequality d+c ν > 2

Conclusions

• even weak disorder can have surprisingly strong effects on a phase transition

• rare regions play a much bigger role at quantum phase transitions than atclassical transitions

• effective dimensionality of rare regions determines character of Griffithssingularities

• in recent years, experimental evidence for quantum Griffiths singularities andsmeared phase transitions has been found at quantum phase transitions in dirtymetals

Quenched disorder at quantum phase transitions leads to a rich varietyof new effects and exotic phenomena

Reviews: T.V., J. Phys. A 39, R143 (2006); J. Low Temp. Phys. 161, 299 (2010)